5 lim(cid:21)BC (cid:15)(cid:21)(cid:31)(cid:22) (cid:15)(cid:21) (cid:8) ) Conversely, if a Fibonacci number is divided by the following Fibonacci number, the result will be close to the reciprocal of phi. Again, the larger the two numbers used, the closer the result will be to the reciprocal of phi (Posamentier & Lehmann, 2007, pp. 107109). (cid:15)(cid:27) (cid:15)(cid:23) (cid:8) 13 21 (cid:8) 0.6190476190 (cid:15)(cid:22)(cid:16) (cid:15)(cid:22)(cid:17) (cid:8) 233 377 (cid:8) 0.6180371353 (cid:15)(cid:22)(cid:24) (cid:15)(cid:30)! (cid:8) 4,181 6,765 (cid:8) 0.6180339985 Fibonacci numbers become even more closely linked to the golden ratio when powers of phi are considered. First, ) (cid:30) is written in terms of ) , which after simplification yields . Each successive power of phi can then be written in terms of factors of (cid:30) ) previous powers of phi. The result of each power is a multiple of (cid:8) ) (cid:11) 1

It ) 18 FIBONACCI SEQUENCE turns out that the coefficient of phi and the constant are consecutive Fibonacci numbers in

113-114). ) (cid:16) (cid:8) ) ) (cid:30) (cid:8) )(cid:28)) (cid:11) 1(cid:29) (cid:8) ) (cid:30) (cid:11) ) (cid:8) (cid:28)) (cid:11) 1(cid:29) (cid:11) ) (cid:8) 2) (cid:11) 1 ) (cid:17) (cid:8) ) (cid:30) ) (cid:30) (cid:8) 3) (cid:11) 2 ) (cid:18) (cid:8) ) (cid:16) ) (cid:30) (cid:8) 5) (cid:11) 3 ) (cid:19) (cid:8) ) (cid:16) ) (cid:16) (cid:8) 8) (cid:11) 5 ) (cid:27) (cid:8) ) (cid:17) ) (cid:16) (cid:8) 13) (cid:11) 8 Golden rectangle. Throughout the course of history, there is a rectangle whose proportions are found most pleasing to the eye. It is neither too fat nor too skinny, neither too long nor too short. People will subconsciously choose this rectangle over another one with different proportions. This rectangle, considered the most perfectly shaped rectangle, is known as the golden rectangle (Garland, 1987, pp. 19-20). This rectangle is

In the late 1800s, Gustav Fechner, a German psychologist, invested a good deal H I (cid:8) of time into researching the subject. He measured thousands of common rectangles, from I H(cid:31)I playing cards and books to windows and writing pads, and he ultimately found that in most of them, the ratio of length to width was close to phi. Fechner also conducted a study in which he asked a large number of people to choose the rectangle out of a group of rectangles was the most pleasing to the eye. His findings showed that the largest percentage of people preferred the rectangle with a ratio of 21:34. These numbers are consecutive Fibonacci numbers, and their ratio approaches the reciprocal of phi. The rectangle most preferred by people was a golden rectangle (Posamentier & Lehmann, 2007, pp. 115-117). 19